In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
and related areas of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the group algebra is any of various constructions to assign to a
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
an
operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study o ...
(or more generally a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
), such that representations of the algebra are related to representations of the group. As such, they are similar to the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
associated to a discrete group.
The algebra ''Cc''(''G'') of continuous functions with compact support
If ''G'' is a
locally compact Hausdorff group, ''G'' carries an essentially unique left-invariant countably additive
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
...
''μ'' called a
Haar measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This Measure (mathematics), measure was introduced by Alfr� ...
. Using the Haar measure, one can define a
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
operation on the space ''C
c''(''G'') of complex-valued continuous functions on ''G'' with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...
; ''C
c''(''G'') can then be given any of various
norms and the
completion will be a group algebra.
To define the convolution operation, let ''f'' and ''g'' be two functions in ''C
c''(''G''). For ''t'' in ''G'', define
:
The fact that
is continuous is immediate from the
dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
. Also
:
where the dot stands for the product in ''G''. ''C
c''(''G'') also has a natural
involution
Involution may refer to: Mathematics
* Involution (mathematics), a function that is its own inverse
* Involution algebra, a *-algebra: a type of algebraic structure
* Involute, a construction in the differential geometry of curves
* Exponentiati ...
defined by:
:
where Δ is the
modular function on ''G''. With this involution, it is a
*-algebra.
Theorem. With the norm:
:
''Cc''(''G'') becomes an involutive normed algebra with an approximate identity.
The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed, if ''V'' is a compact neighborhood of the identity, let ''f
V'' be a non-negative
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
supported in ''V'' such that
:
Then
''V'' is an approximate identity. A group algebra has an identity, as opposed to just an approximate identity,
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the topology on the group is the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
.
Note that for discrete groups, ''C
c''(''G'') is the same thing as the complex group ring C
'G''
The importance of the group algebra is that it captures the
unitary representation
In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in the ca ...
theory of ''G'' as shown in the following
Theorem. Let ''G'' be a locally compact group. If ''U'' is a strongly continuous unitary representation of ''G'' on a Hilbert space ''H'', then
:
is a non-degenerate bounded *-representation of the normed algebra ''Cc''(''G''). The map
:
is a bijection between the set of strongly continuous unitary representations of ''G'' and non-degenerate bounded *-representations of ''Cc''(''G''). This bijection respects unitary equivalence and strong containment. In particular, ''U'' is irreducible if and only if ''U'' is irreducible.
Non-degeneracy of a representation of ''C
c''(''G'') on a Hilbert space ''H''
means that
:
is dense in ''H''
.
The convolution algebra ''L''1(''G'')
It is a standard theorem of
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
that the completion of ''C
c''(''G'') in the ''L''
1(''G'') norm is isomorphic to the space
''L''1(''G'') of equivalence classes of functions which are integrable with respect to the
Haar measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This Measure (mathematics), measure was introduced by Alfr� ...
, where, as usual, two functions are regarded as equivalent if and only if they differ only on a set of Haar measure zero.
Theorem. ''L''1(''G'') is a Banach *-algebra with the convolution product and involution defined above and with the ''L''1 norm. ''L''1(''G'') also has a bounded approximate identity.
The group C*-algebra ''C*''(''G'')
Let C
'G''be the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
of a
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...
''G''.
For a locally compact group ''G'', the group
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...
''C*''(''G'') of ''G'' is defined to be the C*-enveloping algebra of ''L''
1(''G''), i.e. the completion of ''C
c''(''G'') with respect to the largest C*-norm:
:
where ranges over all non-degenerate *-representations of ''C
c''(''G'') on Hilbert spaces. When ''G'' is discrete, it follows from the triangle inequality that, for any such , one has:
:
hence the norm is well-defined.
It follows from the definition that, when G is a discrete group, ''C*''(''G'') has the following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: any *-homomorphism from C
'G''to some B(''H'') (the C*-algebra of
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
s on some
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
''H'') factors through the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota(x)=x.
An inclusion map may also be referred to as an inclu ...
:
:
The reduced group C*-algebra ''Cr*''(''G'')
The reduced group C*-algebra ''C
r*''(''G'') is the completion of ''C
c''(''G'') with respect to the norm
:
where
:
is the ''L''
2 norm. Since the completion of ''C
c''(''G'') with regard to the ''L''
2 norm is a Hilbert space, the ''C
r*'' norm is the norm of the bounded operator acting on ''L''
2(''G'') by convolution with ''f'' and thus a C*-norm.
Equivalently, ''C
r*''(''G'') is the C*-algebra generated by the image of the left
regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation.
One distinguishes the left regular re ...
on ''ℓ''
2(''G'').
In general, ''C
r*''(''G'') is a quotient of ''C*''(''G''). The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if ''G'' is
amenable.
von Neumann algebras associated to groups
The group von Neumann algebra ''W*''(''G'') of ''G'' is the enveloping von Neumann algebra of ''C*''(''G'').
For a discrete group ''G'', we can consider the
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
ℓ
2(''G'') for which ''G'' is an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
. Since ''G'' operates on ℓ
2(''G'') by permuting the basis vectors, we can identify the complex group ring C
'G''with a subalgebra of the algebra of
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
s on ℓ
2(''G''). The weak closure of this subalgebra, ''NG'', is a
von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann al ...
.
The center of ''NG'' can be described in terms of those elements of ''G'' whose
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
is finite. In particular, if the identity element of ''G'' is the only group element with that property (that is, ''G'' has the
infinite conjugacy class property), the center of ''NG'' consists only of complex multiples of the identity.
''NG'' is isomorphic to the
hyperfinite type II1 factor if and only if ''G'' is
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
,
amenable, and has the infinite conjugacy class property.
See also
*
Graph algebra
*
Incidence algebra
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebra#Subalgebras_for_algebras_over_a_ring_or_field, Subalgebras c ...
*
Hecke algebra of a locally compact group
*
Path algebra
*
Groupoid algebra
*
Stereotype algebra
*
Stereotype group algebra
*
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...
Notes
References
*
*
*
*
*
* {{PlanetMath attribution, id=3628, title=Group $C^*$-algebra
Algebras
C*-algebras
von Neumann algebras
Unitary representation theory
Harmonic analysis
Lie groups